NCSA CAII Fall Seminar Series: Machine Learning and Inverse Problems in Scientific Imaging
Dr. Zhizhen Zhao, Assistant Professor in the Department of Electrical and Computer Engineering at the University of Illinois at Urbana-Champaign, will give a presentation during the CAII Seminar Series on Monday, November 1 at 11:00 a.m. The talk is titled “Machine learning and Inverse Problems in Scientific Imaging." Abstract: Machine learning for scientific imaging is a rapidly growing area of research used to characterize physical, material, chemical, and biological processes in scientific experiments. For solving the inverse problems encountered in scientific imaging, we need to incorporate physics-based imaging processes and the geometric properties of the imaging data. In this talk, I will present some of our recent studies in cryo-electron microscopy (EM) image analysis and unknown view tomography. Typical computed tomographic reconstruction assumes that the imaging directions are given, however, there are some situations in which the imaging directions are unknown, for example, when imaging a moving object. In addition, for certain applications, we need to use low dose imaging, which makes the data extremely noisy. I will give a brief review of the manifold learning and distribution learning and show how these perspectives combined with physics-based imaging model may help us to advance unknown view tomography and cryo-EM image analysis. Speaker Bio: Zhizhen Zhao is an assistant professor in the Department of Electrical and Computer Engineering at the University of Illinois, Urbana-Champaign, with affiliation to the Coordinated Science Laboratory and the National Center Supercomputing Applications. She is also an affiliate assistant professor in the Department of Mathematics and the Department of Statistics. Prior to joining ECE Illinois in 2016, she was a Courant Instructor at the Courant Institute of Mathematical Sciences, New York University. She received her PhD in Physics from Princeton University in 2013 working with Amit Singer and graduated from Trinity College, Cambridge University, with bachelor's and master's degrees in physics in 2008. All presentations will be recorded and will available on the CAII website shortly after the presentation. |
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Symplectic and Poisson geometry seminar: Revisiting and extending Poisson-Nijenhuis structures
Poisson-Nijenhuis structures arise in various settings, such as the theory of integrable systems, Poisson-Lie theory and quantization. By revisiting this notion from a new viewpoint, I will show how it can be naturally extended to the realm of Dirac structures, with applications to integration results in (holomorphic) Poisson geometry. |
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Graph Theory and Combinatorics Seminar: Some new results on d-wise, t-intersecting systems
A k-uniform family of sets F is d-wise, t-intersecting if the intersection of any d sets from F contains at least t elements. In this talk, I will state and prove analogues of the classical Erdos-Ko-Rado and Hilton-Milner theorems for d-wise, t-intersecting families, improving upon several earlier results by a number of authors, including O'Neill-Verstraete and Tokushige. I will also discuss some possible directions for future research. Time permitting, I will indicate how d-wise, t-intersecting families can be used to construct K_{s,t}-intersecting families of size larger than the trivial construction, answering a question of Ellis. This is joint work with Jozsef Balogh. |
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Harmonic Analysis and Differential Equations Seminar: Floquet Hamiltonians - effective gaps and resonant decay
Floquet topological insulators are an emerging category of materials whose properties are transformed by time-periodic forcing. Can their properties be understood from their first-principles continuum models, i.e., from a driven Schrodinger equation? First, we study the transformation of graphene from a conductor into an insulator under a time-periodic magnetic potential. We show that the dynamics of certain wave-packets are governed by a Dirac equation, which has a spectral gap property. This gap is then carried back to the original Schrodinger equation in the form of an “effective gap” - a new and physically-relevant relaxation of a spectral gap. Next, we consider periodic media with a localized defect, and ask whether edge/defect modes remain stable under forcing. In a model of an optical waveguides array, we see how such modes decay and disappear due to resonant coupling with the radiation modes. |
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Cluster Algebra/IRT Seminar: Dimers and circle patterns
The dimer model is a model from statistical mechanics corresponding to random perfect matchings on graphs. Circle patterns are a class of embeddings of planar graphs such that every face admits a circumcircle. We describe how to construct a 't-embedding' (or a circle pattern) of a dimer planar graph using its Kasteleyn weights, and discuss algebro-geometric properties of these embeddings. This new class of embeddings is the key for studying Miquel dynamics, a discrete integrable system on circle patterns: we identify Miquel dynamics on the space of square-grid circle patterns with the Goncharov-Kenyon dimer dynamics and deduce the integrability of the former one and show that the evolution is governed by cluster algebra mutations. |
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Teaching & Diversity Seminar: Avoiding the Academic Savior Complex and Building Better Mentoring Relationships.
Dr. Harris will talk about Avoiding the Academic Savior Complex and Building Better Mentoring Relationships. In this talk, Dr. Harris will discuss some mentoring experiences and their effects on her mathematical self-confidence and career progression. She will then share concrete ideas on how to build better mentoring relationships. Dr. Pamela E. Harris is a Mexican-American mathematician and serves as Associate Professor in the Department of Mathematics and Statistics and Faculty Fellow of the Davis Center and the Office of Institutional Diversity, Equity, and Inclusion at Williams College. Dr. Pamela E. Harris's research is in algebraic combinatorics and she is the author of over 50 peer-reviewed research articles in internationally recognized journals. Her professional mission is to develop learning communities that reinforce students’ self-identity as scientists, in particular for women and underrepresented minorities. In order to provide visibility to and increase the positive impact of the role models within our community, Dr. Harris co-founded Lathisms.org, a platform that features the contributions of Latinx and Hispanic scholars in the Mathematical Sciences. She cohosts the podcast Mathematically Uncensored and has recently coauthored the books Asked And Answered: Dialogues On Advocating For Students of Color in Mathematics and Practices and Policies: Advocating for Students of Color in Mathematics. |
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Combinatorics Colloquium: Algebraic constructions in combinatorics
There are two main sources to produce non trivial combinatorial structures. One can use probability theory for typical cases and a bit of algebra for symmetric structures. Here we briefly review classical and new developments and also give examples on how to combine these two powerful methods. The talk is targeting general mathematicians, with little combinatorics backgrounds. |
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Algebra, Geometry & Combinatorics: Newell-Littlewood numbers
The Newell-Littlewood numbers are defined in terms of the Littlewood-Richardson coefficients from algebraic combinatorics. Both appear in representation theory as tensor product multiplicities for a classical Lie group. This talk concerns the question, Which multiplicities are nonzero? In 1998, Klyachko established common linear inequalities defining both the eigencone for sums of Hermitian matrices and the saturated Littlewood-Richardson cone. We prove some analogues of Klyachko's nonvanishing results for the Newell-Littlewood numbers. This is joint work with Shiliang Gao (UIUC), Gidon Orelowitz (UIUC), and Nicolas Ressayre (Universite Claude Bernard Lyon I). The presentation is based on arXiv:2005.09012, arXiv:2009.09904, and arXiv:2107.03152. |
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2021 Department of Mathematics Fall Faculty Meeting
The 2021 Fall Faculty Meeting will be held virtually via Zoom. |
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Undergraduate Friday Seminar: An Introduction to Geometry in Physics
Christopher Gottardi-Littell! Chris is a junior undergraduate in the Math program, with interests primarily in mathematical physics. And on Friday, he’ll be here to talk about some of the interesting geometric settings we use for physical models: An Introduction to Geometry in Physics Since Newton’s breakthrough, mathematicians have examined classical mechanics in terms of modern theories of space. The result is a series of mathematical pictures of the world we see. I will introduce the Lagrangian and Hamiltonian formulations of mechanics, with emphasis on the geometric structures in which they are phrased. |
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The symplectic (A-infinity,2)-category and a simplicial version of the 2D Fulton-MacPherson operad
The symplectic (A-infinity,2)-category Symp, which is currently under construction by myself and my collaborators, is a 2-category-like structure whose objects are symplectic manifolds and where hom(M,N) := Fuk(M^- x N). Symp is a coherent algebraic structure which encodes the functoriality properties of the Fukaya category. This talk will begin with the following question: what can we say about the part of Symp that knows only about a single symplectic manifold M, and the diagonal Lagrangian correspondence from M to itself? We expect that the answer to this question should be a chain-level algebraic structure on symplectic cohomology, and in this talk I will present progress toward confirming this. Specifically, I will present a "simplicial version" of the 2-dimensional Fulton-MacPherson operad, which may be of independent topological interest. If there is time, I will also explain how this development can be used to give a definition of (A-infinity,2)-categories that involves only finitely many operations of each arity. |
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Topology seminar: Chromatic homotopy theory via spectral algebraic geometry
A key aspect of chromatic homotopy theory is that structural properties of the stable homotopy category are reflected in the algebro-geometric properties of the moduli stack of formal groups. In this talk, we will discuss how to make that connection precise in the context of non-connective spectral algebraic geometry, using the stack of oriented formal groups. |
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Graph Theory and Combinatorics Seminar: Uniform Orderings for Generalized Coloring Numbers
The generalized coloring numbers $\co_{r}(G)$ (also denoted by $\sco_r(G)$) nd $\wc_r(G)$ of a graph~$G$ are generalizations of the usual coloring number that have found important theoretical and algorithmic applications. For each distance~$r$, these numbers are determined by an ``optimal'' ordering of the vertices of~$G$. We study the question of whether it is possible to find a single "uniform'' ordering that is "good'' for all distances~$r$. We show that the answer to this question is essentially yes. Our results give new characterizations of graph classes with bounded expansion and nowhere dense graph classes. This is joint work with Jan van den Heuvel. |
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Probability Seminar: A tiling proof of the pointwise ergodic theorem for actions of amenable groups along Tempelman F{\o}lner sequences
A pointwise ergodic theorem for the action of a countable group $\Gamma$ on a probability space equates the global ergodicity of the action to its pointwise combinatorics. Our main result is a short, combinatorial proof of the pointwise ergodic theorem for actions of amenable groups along Tempelman F{\o}lner sequences, which is a slightly less general version of Lindenstrauss's celebrated theorem. We will discuss a very short proof (due to Tserunyan) of Birkhoff's classical pointwise ergodic theorem, and, using this proof as an outline, we will sketch the proof of the theorem for Tempelman F{\o}lner sequences. This is joint work with Jon Boretsky. |
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Actuarial Science and Financial Mathematics Seminar: How does global climate change influence regional extremes and what are the decision-relevant uncertainties?
Abstract: Earth is warming, and the damages associated with climate and weather extremes (droughts, heatwaves, floods, hurricanes) are increasing. Projecting these changes into the future is difficult due to: incomplete understanding of the physical processes, inadequate numerical models and resolution, and relatively short observational records. Here we highlight some of the current grand climate change problems, in particular connecting global changes to fine-scale impacts, and we present some recent work analyzing how climate and weather extremes are changing in response to global warming. Ryan L. Sriver is an associate professor of atmospheric sciences at the University of Illinois at Urbana-Champaign. Prior to joining UIUC in 2012, he worked as a research associate in Penn State's Department of Geosciences and as a NOAA Climate and Global Change postdoctoral fellow in Penn State's Department of Meteorology. Ryan graduated from Purdue University with a PhD in Earth and Atmospheric Sciences. Ryan's research seeks to develop a deeper understanding about the physical processes influencing variability within Earth's climate system and to quantify relevant uncertainties surrounding future climate projections. His work combines observational products, statistical methods and tools, and numerical models spanning a wide range of complexities and scales to understand how extreme weather and climate events are changing with global warming, what are the physical drivers, and what are the implications for natural and human systems. His research interests include: Climate Dynamics, Earth System Modeling, Ocean-Atmosphere Interactions, Uncertainty and Risk, Weather and Climate Extremes, Tropical Cyclones, Sea-Level Rise, Seasonal Prediction, and Multi-Sector Dynamics. |
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Graduate Student Colloquium: The distribution of reduced quadratic irrationals arising from continued fraction expansions.
It is known that the reduced quadratic irrationals arising from regular continued fraction expansions are uniformly distributed when ordered by their length with respect to the Gauss measure. In this talk, I will describe a number theoretical approach developed by Kallies, Ozluk, Peter and Snyder, and then by Boca and Ustinov, that gives the error in the asymptotic behavior of this distribution. Moreover, I will present the respective result for the distribution of reduced quadratic irrationals that arise from even (joint work with F. Boca) and odd continued fractions. No prior knowledge about continued fractions is needed to attend the talk. |
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Student Cluster Algebra Seminar
This seminar will consist of the second portion of the following talk: Title: A brief introduction to cluster scattering diagrams Abstract: We will begin by defining scattering diagrams, which were first introduced in two dimensions by Kontsevich and Soibelman and then in arbitrary dimension by Gross and Siebert as a tool for constructing mirror spaces. Using concrete examples, we will then walk through the cluster scattering diagram construction given by Gross, Hacking, Keel, and Kontsevich and give definitions of cluster varieties, broken lines, theta functions, and other relevant objects. If time allows, we will then briefly sketch how cluster scattering diagrams are used to prove important results for ordinary cluster algebras, including positivity and the existence of the theta basis. |
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Chaeryn Lee: preliminary exam
Preliminary Exam of Chaeryn Lee, PhD student, Department of Mathematics. |
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Food For Thought: Math Faculty Speak at CAS
11am, Steven Bradlow, Geometric Structures on Pretzels and Other Surfaces: What Goes on in a Mathematics Research Institute As a CAS Fellow, Professor Bradlow spent three months in Fall 2019 at the Mathematical Sciences Research Institute, a remarkable place at the top of the Berkeley hills overlooking the San Francisco Bay. He was there for a program devoted to the mathematics of surfaces, a subject where geometry is no longer governed by Euclid’s postulates and exotic structures play a role in questions as varied as how to find the best way to avoid the corners of a polyhedron and questions in quantum field theories in physics. He will describe some of the themes of the program but will leaven the mathematics with some digressions about the hosting institute and its history. Noon, Vesna Stojanoska, Solving Polynomial Equations with Homotopy Theory Given a general polynomial, it can be hard or impossible to solve it, or even to tell if solutions exist within a fixed number system. The set of solutions is rather rigid: numbers that are “close” to a solution usually are not themselves solutions. So how can homotopy theory, the study of shapes that can be continuously deformed without changing their nature, help in solving polynomials? Professor Stojanoska will sketch some old ideas and new developments in the world of homotopical approaches to arithmetic questions. |
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Mathematics Colloquium: Three Open Questions in the Three-Body Problem
The classical 3-body problem is alive and well. I start with a pictorial survey of solutions, then describe two or three open questions and recent progress on them. The shape sphere, a sphere parameterizing oriented similarity classes of planar triangles, will provide a unifying perspective. |
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Functional & geometric Inequalties: An Introduction
Abstract: This talk introduces functional and geometric inequalities, opening with a discussion about what they are and key questions that frame their study. We prove the Polya-Szego inequality and then use it to establish the classical Faber-Krahn inequality, which states that balls minimize the first eigenvalue of the Dirichlet Laplacian among all sets of the same volume. We conclude by outlining what a more general Faber-Krahn inequality looks like and how one might go about deriving it. This talk is based on a paper by Brasco and De Phillips (2016). |
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Graduate Student Homotopy Theory Seminar: ∞-categorical Kummer theory.
Kummer theory provides a way of recognizing the isomorphism classes of (certain) Galois extensions of fields containing enough roots of unity. Running a similar machinery for general classical commutative rings leads to the classification of (certain) Galois extensions of the ring which involves the Picard spectrum of the ring. Schlank, et al. have designed a version of this theory that works for nice presentable additive symmetric monoidal ∞-categories which involves the Picard spectrum of this category. I will go over their version of the theory in this talk. |
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Undergraduate Friday Seminar: IGL Info Session
This week, on Friday Nov. 12th at 4pm in room 245 of Altgeld Hall, we'll be joined by Madie Farris! Madie is a PhD student here in the Math program and also the Research Manager for the Illinois Geometry Lab. On Friday, they'll be joining us to give some helpful info about the IGL - tips for applying, potential future research projects, and more. The IGL aims to get undergraduates at U of I involved in math research and outreach. During this talk I will outline the different ways you can get involved with the IGL, tips and tricks for applications, give an overview of some past and future research projects, and answer any questions you might have about our programs. If you've ever been interested in math outreach or research, then this is a great opportunity to hear about ways to get started! |
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